Definite Integration
Integral Equation — Finding e+f(0)
nta_pyq_2026_jan
Grade 12
Question:
If $f(x)$ satisfies the relation $f(x)=e^x+\displaystyle\int_0^1(y+xe^y)f(y)\,dy$, then $e+f(0)$ is equal to _____
Step-by-Step Solution
Key Concept: Let $A=\int_0^1 f(y)\,dy$ and $B=\int_0^1 ye^y f(y)\,dy$. Then $f(x)=(1+B)e^x+Ax$. Substitute back and integrate to find $A$ and $B$.
$f(0)=1-e$. $e+f(0)=2$.
Correct Answer: 2